The ordinary-double-point volume gap

What OpenAI claims

Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most 2(n-1)ⁿ. Equality holds precisely for an analytic ordinary double point.

Papers (2)

1. The ordinary-double-point gap in every dimension

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most 2(n-1)ⁿ, with equality precisely at an analytic ordinary double point.

2. The normalized-volume gap in dimension four

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.